This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.


Enriched category theory

Limits and colimits



In a closed symmetric monoidal category VV the internal hom [,]:V op×VV[-,-] : V^{op} \times V \to V satisfies the natural isomorphism

[v 1,[v 2,v 3]][v 2,[v 1,v 3]] [v_1,[v_2,v_3]] \simeq [v_2,[v_1,v_3]]

for all objects v iVv_i \in V (prop.). If we regard VV as a VV-enriched category we write V(v 1,v 2):=[v 1,v 2]V(v_1,v_2) := [v_1,v_2] and this reads

V(v 1,V(v 2,v 3))V(v 2,V(v 1,v 3)). V(v_1,V(v_2,v_3)) \simeq V(v_2,V(v_1,v_3)) \,.

If we now pass more generally to any VV-enriched category CC then we still have the enriched hom object functor C(,):C op×CVC(-,-) : C^{op} \times C \to V. One says that CC is powered over VV if it is in addition equipped also with a mixed operation :V op×CC\pitchfork : V^{op} \times C \to C such that (v,c)\pitchfork(v,c) behaves as if it were a hom of the object vVv \in V into the object cCc \in C in that it satisfies the natural isomorphism

C(c 1,(v,c 2))V(v,C(c 1,c 2)). C(c_1,\pitchfork(v,c_2)) \simeq V(v,C(c_1,c_2)) \,.



Let VV be a closed monoidal category. In a VV-enriched category CC, the power of an object yCy\in C by an object vVv\in V is an object (v,y)C\pitchfork(v,y) \in C with a natural isomorphism

C(x,(v,y))V(v,C(x,y)) C(x, \pitchfork(v,y)) \cong V(v, C(x,y))

where C(,)C(-,-) is the VV-valued hom of CC and V(,)V(-,-) is the internal hom of VV.

We say that CC is powered or cotensored over VV if all such power objects exist.


Powers are frequently called cotensors and a VV-category having all powers is called cotensored, while the word “power” is reserved for the case V=V= Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.


  • Powers are a special sort of weighted limits. Conversely, all weighted limits can be constructed from powers together with conical limits. The dual colimit notion of a power is a copower.


  • VV itself is always powered over itself, with (v 1,v 2):=[v 1,v 2]\pitchfork(v_1,v_2) := [v_1,v_2].

  • Every locally small category CC (V=(Set,×)V = (Set,\times) ) with all products is powered over Set: the powering operation

    (S,c):= sSc \pitchfork(S,c) := \prod_{s\in S} c

    of an object cc by a set SS forms the |S||S|-fold cartesian product of cc with itself, where |S||S| is the cardinality of SS.

    The defining natural isomorphism

    Hom C(c 1,(S,c 2))Hom Set(S,Hom C(c 1,c 2)) Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))

    is effectively the definition of the product (see limit).


Last revised on July 19, 2018 at 08:35:31. See the history of this page for a list of all contributions to it.